L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.866 − 0.5i)9-s + (−0.366 + 1.36i)11-s + (1 + i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.499i)18-s + (0.366 + 1.36i)19-s + (−0.999 − i)22-s + (0.5 − 0.866i)23-s + (−1.36 + 0.366i)26-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.866 − 0.5i)9-s + (−0.366 + 1.36i)11-s + (1 + i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.499i)18-s + (0.366 + 1.36i)19-s + (−0.999 − i)22-s + (0.5 − 0.866i)23-s + (−1.36 + 0.366i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9117866991\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9117866991\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1 + i)T - iT^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.079781307779616403075066403958, −8.163547567844203383280077390085, −7.71653593580111225470255863986, −6.72285436386458576052302664041, −6.40642917189336291928164042297, −5.26206802988408828164521701473, −4.54924697690797611476917527625, −3.81255980843452818882239453918, −2.24129954787090119875489166080, −1.05999592554804088225789017277,
0.901237921274665291271228206432, 2.30223672879266461890622861597, 2.98791582938490826594922992975, 3.75196618953381607216766027045, 5.17559268188793591097824518282, 5.48726647013841131235449798065, 6.57123379641762867296971696037, 7.85985442790228357043241383399, 8.385184604166063445488922893632, 8.711081618000335813638954370291